Symmetries and monotones in Markovian quantum dynamics

TL;DR

This paper introduces a method to derive scalar monotonic functions from symmetries of Markovian quantum dynamics, which help identify reachable states and generalize Noether's theorem in quantum systems.

Contribution

It presents a novel construction linking symmetries of quantum generators to monotones, extending the understanding of constraints in Markovian quantum evolution.

Findings

Constructed scalar monotones from generator symmetries.

Applied the framework to dephasing and Davies generators.

Provided insights into constraints on quantum state evolution.

Abstract

What can one infer about the dynamical evolution of quantum systems just by symmetry considerations? For Markovian dynamics in finite dimensions, we present a simple construction that assigns to each symmetry of the generator a family of scalar functions over quantum states that are monotonic under the time evolution. The aforementioned monotones can be utilized to identify states that are non-reachable from an initial state by the time evolution and include all constraints imposed by conserved quantities, providing a generalization of Noether's theorem for this class of dynamics. As a special case, the generator itself can be considered a symmetry, resulting in non-trivial constraints over the time evolution, even if all conserved quantities trivialize. The construction utilizes tools from quantum information-geometry, mainly the theory of monotone Riemannian metrics. We analyze the prototypical cases of dephasing and Davies generators.